Matter-Domination Calculator

About the Matter-Domination Calculator

The Matter-Domination Calculator is an advanced, scientifically accurate online tool designed to determine the precise cosmic epoch during which non-relativistic matter (cold dark matter + baryons) dominated the energy content of the universe. This period, known as the matter-dominated era, is a cornerstone of modern physical cosmology within the standard ΛCDM model.

What is Matter Domination in Cosmology?

In the standard cosmological model, the universe has gone through three major phases of expansion dominance: radiation domination (very early universe), matter domination (from ~50,000 years until ~9.8 billion years after the Big Bang), and the current dark energy (Λ) domination. The transition from radiation to matter domination occurred at the equality redshift zₑq when the energy density of matter equaled that of radiation.

Matter domination is crucial because this is when cosmic structure formation began in earnest. Density perturbations could grow linearly via gravitational instability during this phase, leading to the formation of galaxies, clusters, and the large-scale structure we observe today.

Scientific Foundation and Formulas Used

This Matter-Domination Calculator uses the exact solutions of the Friedmann equation in a flat ΛCDM universe:

H²(a) / H₀² = Ωᵣ,₀ a⁻⁴ + Ωₘ,₀ a⁻³ + ΩΛ,₀

The matter-radiation equality occurs when:

Ωₘ,�0 a⁻³ = Ωᵣ,₀ a⁻⁴

→ aₑq = Ωᵣ,₀ / Ωₘ,₀

→ zₑq = (Ωₘ,₀ / Ωᵣ,₀) − 1 ≈ 3390 for Planck 2018 values

The transition to dark energy domination occurs when Ωₘ(a) = ΩΛ(a), giving:

aΛ = (Ωₘ,₀ / ΩΛ,₀)¹ᐟ³

zΛ ≈ 0.3–0.4 depending on parameters

The age of the universe at any scale factor is computed via the integral:

t(a) = ∫₀ᵃ da' / [a' H(a')]

When Did Matter Domination Begin and End?

Using Planck 2018 cosmological parameters (Ωₘ,₀ ≈ 0.315, h ≈ 0.674), the matter-dominated era lasted from:

• Start: z ≈ 3390 (about 47,000 years after the Big Bang)
• End: z ≈ 0.38 (about 9.8 billion years after the Big Bang)

Why Use This Matter-Domination Calculator?

Researchers, students, and cosmology enthusiasts use this tool to:

• Understand structure formation timelines
• Compare different cosmological models
• Study the impact of varying Ωₘ,₀ or h on cosmic history
• Teach physical cosmology and ΛCDM model
• Visualize how close we are to the end of matter domination

User Guidelines

1. Enter current matter density Ωₘ,₀ (best value: 0.315 ± 0.007 from Planck)

2. Leave radiation field empty for automatic calculation (Ωᵣ,₀ ≈ 4.15×10⁻⁵ h⁻²)

3. Adjust h if using different datasets (e.g., SH0ES gives h ≈ 0.74)

4. Click "Calculate" to get exact redshifts and cosmic ages

This calculator assumes a flat universe (Ωₖ=0) and uses the latest radiation content from CMB measurements.

Importance of the Matter-Dominated Era

Without a prolonged matter-dominated phase, galaxies as we know them would not exist. During radiation domination, pressure from photons prevented gravitational collapse. Only after matter-radiation equality could overdensities grow significantly, eventually forming stars and galaxies.

The duration and timing of matter domination directly affect the power spectrum of cosmic structures, the number of galaxy clusters, and the reionization history.

References & Further Reading

Planck Collaboration (2020), Dodelson (2020) Modern Cosmology, Ryden "Introduction to Cosmology", Weinberg "Cosmology".